Deformed Coset Models From Gauged WZW Actions

A general Lagrangian formulation of integrably deformed G/H-coset models is given. We consider the G/H-coset model in terms of the gauged Wess-Zumino-Witten action and obtain an integrable deformation by adding a potential energy term Tr(gTgT̄ ), where algebra elements T, T̄ belong to the center of the algebra h associated with the subgroup H. We show that the classical equation of motion of the deformed coset model can be identified with the integrability condition of certain linear equations which makes the use of the inverse scattering method possible. Using the linear equation, we give a systematic way to construct infinitely many conserved currents as well as soliton solutions. In the case of the parafermionic SU(2)/U(1)-coset model, we derive n-solitons and conserved currents explicitly. 1 E-mail address; [email protected] Integrable field theories in two dimensions (2-d IFT) have been successfully developed in the past decades. Recently, Zamolodchikov has shown that there exist deformations of conformal field theories (2-d CFT) with relevant operators which preserve integrability[1]. In the case of certain rational conformal field theories, these deformations have been explained in terms of the affine extension of the Toda field theory[2][3]. However, a general Lagrangian framework for integrably deformed 2-d CFT’s has not been known so far. The purpose of this letter is to provide a Lagrangian formulation of general G/H-coset models and their integrable deformations in the context of the gauged Wess-Zumino-Witten (WZW) model[4][5]. We show that an integrable deformation of general G/H-coset models is possible when the gauged WZW action for the G/H-coset model is added by a potential energy term Tr(gTgT̄ ), where algebra elements T, T̄ belong to the center of the algebra h associated with the subgroup H. In the case of SU(2)/U(1), this reduces to the Lagrangian, given recently by Bakas[6], of the parafermion model deformed by the energy operator Φ1,3. The main observation of this work is that the classical equation of motion of the deformed coset model takes the form of a zero curvature which can be identified with the integrability condition of the associated linear equations with a spectral parameter. This allows us to apply the inverse scattering method to the problem and using this method, we give a systematic way to construct infinitely many conserved currents and n-soliton solutions. As an example, we construct explicitly conserved currents and n-soliton solutions of the deformed parafermionic SU(2)/U(1)-coset model. We first recall that a Lagrangian of the G/H-coset model is given in terms of the gauged WZW functional[4][5], which in light-cone variables is I(g, A, Ā) = IWZW (g) + 1 2π ∫ Tr(−A∂̄gg + Āg∂g + AgĀg −AĀ) (1) where IWZW (g) is the usual WZW action [5] for a map g : M →G on two-dimensional Minkowski space M . The connection A, Ā gauge the anomaly free subgroup H of G. In this letter, we take the diagonal embedding of H in GL ×GR, where GL and GR denote left and right group actions by multiplication (g → gLgg R ), so that Eq.(1) becomes invariant under the vector gauge transformation (g → hgh with h : M →H). The key observation to be made is that the equation of motion of the gauged WZW action takes the form of a zero